Court decisions such as Heller have upheld "reasonable restrictions" on rights. The legal question really isn't if the AWB would be a violation of our civil rights, but whether or not it is unreasonable.
One could argue, and many have, that ANY infringement of the right to keep and bear arms is an infringement, but wholly support jailed criminals not having guns (an obvious infringement). If a right can't be infringed, then how can it be lost? Most agree that private citizens should not be allowed to own arms such as bombs and mines. There seems to be a large consensus that mentally unstable or handicapped people should not be allowed to bear arms and there are laws to support this. These all fall under the guise of reasonable restrictions. If you support any such restrictions in any way, then you support the concept of reasonable restrictions.
To argue the AWB is a violation of 2nd Amendment rights given how the courts perceive the 2nd Amendment, then the argument likely would need to be that it is an unreasonable restriction as there are already numerous restrictions in place deemed to be 'reasonable' and surprisingly supported by society as a whole, not just the courts.
"If you look through your scope and see your shoe, aim higher."
-- said to me by my 11 year old daughter before going out for hogs 8/13/2011